Phase shift oscillators are well known in the art. The frequency of the oscillator is typically stabilized by a resistance capacitance ladder which gives the necessary delay (phase shift) in a feedback loop. It is also known that the conventional phase shift oscillators require at least three stages in order to sustain oscillation.
Thus, in a circuit including a pair of operational amplifier integrators of equal gain and a sign changer electrically connected in cascade with gain control between stages, sustained oscillation cannot be maintained. Circuit analysis of such a circuit where e.sub.1 is the voltage into the first integrator and e.sub.2 is the voltage into the second integrator provides: ##EQU1## where .pi. = RC, where R is the resistance between the input voltage e.sub.1 or e.sub.2 and the operational amplifier, C is the capacitance of the negative feedback loop of each amplifier connected from the output to between resistance R and the amplifier input; A = operational amplifier gain; ##EQU2## AND .alpha. AND .beta. ARE THE TRANSMISSIONS OF ATTENUATORS IN THE OUTPUTS OF EACH INTEGRATOR.
Equations (1) and (2), which define a,b,c, and d, follow from the equation for the basic integrator circuit: ##EQU3## since an ideal integrator should have an output given by ##EQU4## the entire second term, ##EQU5## of equation (3) and the A.sup.-.sup.1 of the first term represent sources of error. Both of these errors, however, approach zero as A becomes infinite.
For a positive finite A, the constants a,b,c and d in equations (1) and (2) are all positive. The equations also assure that the sign changer is ideal, that is, e.sub.3 = -e.sub.1 which can be attained by making ##EQU6##
Under these conditions, the solutions of equations (1) and (2) are given by: ##EQU7## where m = [(b + c).sup.2 - 4ad].sup.1/2 and e.sub.10 and e.sub.20 are the initial values of e.sub.1 and e.sub.2 at t = 0. In order for e.sub.1 and e.sub.2 to be periodic it is necessary that m.sup.2 be negative and that b + c = 0; and, for sustained sinusoidal oscillation it is necessary that b = c = 0. However, for a finite A, b and c cannot be equal to zero since each is equala to ##EQU8## Accordingly, a circuit having a pair of integrator circuits as described above and a sign changer in cascade, cannot be made to give sustained oscillations where A is finite. Any initial value of voltages e.sub.10 and e.sub.20 results in a damped oscillation where the damping time constant is .theta. and is equal to 2/(b + c) = .pi.(1 + A). The number of cycles during the time constant is ##EQU9## and since .alpha. and .beta. are less than one, the number of cycles during the damping time constant is less than A/2.pi..
A further problem in oscillator control is that the sensor means, typically a thermistor, used for amplitude control is activated not by the amplitude of the output but rather by the instantaneous output voltage. Thus, at low frequencies, the sensor follows the cyclic variations of the output providing waveform distortions. This behavior normally limits the operating frequency to a value high compared to the reciprocal of the sensor response time.
Accordingly, it is an object of the present invention to provide an oscillator circuit having a pair of integrators in which the gain appears to be infinite in order to provide sustained oscillation in the oscillator circuit. A further object of the present invention is to provide amplitude control without waveform distortion to permit low-frequency operation at frequencies down to and including zero.